Fixed points and braids. II
A basic problem in fixed point theory is to find the least number of fixed points for a homotopy class of maps. That is, given a map \(f: X\to X\) of a connected compact polyhedron X into itself, to find \(MF[f]:=Min\{\#Fix(g)|\) \(g\simeq f: X\to X\}\). The invariant central to this theory is the Nielsen number N(f), defined as the number of essential fixed point classes. N(f) is always a lower bound to MF[f] and it has been a long-standing problem to prove or to disprove that \(N(f)=MF[f]\). We know [the author, Am. J. Math. 102, 749-763 (1980; Zbl 0455.55001)] that if X has no local cut points and X is not a surface of negative Euler characteristic, then \(N(f)=MF[f]\) for every map \(f: X\to X\). The aim of the present paper is to generalize the result of the author [Invent. Math. 75, 69-74 (1984; Zbl 0565.55005)] and to show Theorem 1. Let M be a connected compact surface with Euler characteristic \(\chi (M)<0\). Then there is a map \(f: M\to M\) such that \(N(f)=0<MF[f].\) From this immediately follows the next theorem. \textit{G.-h. Shí} [Acta Math. Sin. 18, 192-202 (1975; Zbl 0375.57004)] has shown that a local cut point is often a source of extra fixed points. In the absence of such local anomaly we are able to characterize those polyhedra for which the equality \(N(f)=MF[f]\) always holds. Theorem 2. Let X be a connected compact polyhedron with no local cut points. Then \(N(f)=MF[f]\) for every map \(f: X\to X\) if and only if X is not a surface of negative Euler characteristic. \(\{\) The results of this article had been announced by the author in Adv. Math., Beijing 14, 63-64 (1985; Zbl 0576.55001).\(\}\).
surface of negative Euler characteristic
scientific article; zbMATH DE number 4000982